Further Results on the Dilogarithm Integral

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2011, Article ID 421601, 10 pages doi:10.1155/2011/421601

Research Article Further Results on the Dilogarithm Integral Biljana Jolevska-Tuneska1 and Brian Fisher2 1

Faculty of Electrical Engineering and Informational Technologies, Ss. Cyril and Methodius University in Skopje, Karpos II bb, 1000 Skopje, Macedonia 2 Department of Mathematics, University of Leicester, Leicester LE1 7RH, England, UK Correspondence should be addressed to Biljana Jolevska-Tuneska, [email protected] Received 5 May 2011; Accepted 4 July 2011 Academic Editor: Ch Tsitouras Copyright q 2011 B. Jolevska-Tuneska and B. Fisher. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The dilogarithm integral Lixs and its associated functions Li xs and Li− xs are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of these functions and other functions are then found.

1. Introduction The dilogarithm integral Lix is defined by Lix −

x 0

ln|1 − t| dt t

1.1

see 1. More generally, we have Lix − s

xs 0

ln|1 − t| dt −s t

x 0

ln|1 − ts | dt t

1.2

for s 1, 2, . . . . The associated functions Li xs and Li− xs are defined by Li xs HxLixs ,

Li− xs H−xLixs Lixs − Li xs ,

where Hx denotes Heaviside’s function.

1.3

2

Journal of Applied Mathematics Next, we define the distribution ln |1 − xs |x−1 by ln |1 − xs |x−1 −s−1 Lixs ,

1.4

and its associated distributions ln |1 − xs |x−1 and ln |1 − xs |x−−1 are defined by ln|1 − xs |x−1 Hx ln|1 − xs |x−1 −s−1 Li xs , ln|1 − xs |x−−1 H−x ln|1 − xs |x−1 −s−1 Li− x .

1.5

The classical definition of the convolution product of two functions f and g is as follows. Definition 1.1. Let f and g be functions. Then the convolution f ∗ g is defined by

f ∗ g x

∞ −∞

ftgx − tdt

1.6

for all points x for which the integral exist. It follows easily from the definition that if f ∗ g exists then g ∗ f exists and f ∗ g g ∗ f,

1.7

and if f ∗ g and f ∗ g or f ∗ g exists, then

f ∗g

f ∗ g

or f ∗ g .

1.8

Definition 1.1 can be extended to define the convolution f ∗ g of two distributions f and g in D with the following definition; see Gel’fand and Shilov 2. Definition 1.2. Let f and g be distributions in D . Then the convolution f ∗ g is defined by the equation

f ∗ g x, ϕ f y , gx, ϕ x y

1.9

for arbitrary ϕ in D, provided f and g satisfy either of the following conditions: a either f or g has bounded support, b the supports of f and g are bounded on the same side. It follows that if the convolution f ∗ g exists by this definition then 1.7 and 1.8 are satisfied.

Journal of Applied Mathematics

3

In order to extend Definition 1.2 to distributions which do not satisfy conditions a or b, let τ be a function in D, see 3, satisfying the conditions: i τx τ−x, ii 0 ≤ τx ≤ 1, iii τx 1 for |x| ≤ 1/2, iv τx 0 for |x| ≥ 1. The function τn is then defined by ⎧ ⎪ 1, ⎪ ⎪ ⎨ τn x τ nn x − nn1 , ⎪ ⎪ ⎪ ⎩ n τ n x nn1 ,

|x| ≤ n, x > n,

1.10

x < −n,

for n 1, 2, . . . . The following definition of the noncommutative neutrix convolution was given in 4. Definition 1.3. Let f and g be distributions in D , and let fn fτn for n 1, 2, . . . . Then the noncommutative neutrix convolution f g is defined as the neutrix limit of the sequence {fn ∗ g}, provided the limit h exists in the sense that N − lim fn ∗ g, ϕ h, ϕ n→∞

1.11

for all ϕ in D, where N is the neutrix, see van der Corput [5], having domain N the positive reals and range N the real numbers, with negligible functions finite linear sums of the functions nλ lnr−1 n,

lnr n :

λ > 0, r 1, 2, . . .

1.12

and all functions which converge to zero in the normal sense as n tends to infinity. In particular, if lim fn ∗ g, ϕ h, ϕ

n→∞

1.13

exists, we say that the non-commutative convolution f g exists. It is easily seen that any results proved with the original definition of the convolution hold with the new definition of the neutrix convolution. Note also that because of the lack of symmetry in the definition of f g the neutrix convolution is in general non-commutative. The following results proved in 4 hold, first showing that the neutrix convolution is a generalization of the convolution. Theorem 1.4. Let f and g be distributions in D , satisfying either condition (a) or condition (b) of Gel’fand and Shilov’s definition. Then the neutrix convolution f g exists and f g f ∗ g.

1.14

4


Theorem 1.5. Let f and g be distributions in D and suppose that the neutrix convolution f g exists. Then the neutrix convolution f g exists and

f g

f g.

1.15

Note however that f g is not necessarily equal to f g but we do have the following theorem. Theorem 1.6. Let f and g be distributions in D and suppose that the neutrix convolution f g exists. If N − limn → ∞ fτn ∗ g, ϕ exists and equals h, ϕ for all ϕ in D, then f g exists and

f g

f g h.

1.16

2. Main Result We define the function Is,r x by Is,r x

x

ur ln|1 − us | du

2.1

0

for r 0, 1, 2, . . . and s 1, 2, . . . . In particular, we define the function Ir x by Ir x I1,r x

2.2

for r 0, 1, 2, . . . . The following theorem was proved in 6. Theorem 2.1. The convolutions Li x ∗ xr and ln |1 − x|x−1 ∗ xr exist and Li x ∗

xr

r r1 1 1 r1 x Li x −1r−i Ir−i xxi r 1 i0 r 1 i

2.3

for r 0, 1, 2, . . . and ln|1 −

x|x−1

∗

xr

r−1 r i0

i

−1r−i Ir−i−1 xxi − Li xxr

2.4

for r 1, 2, . . . . We now prove the following generalization of Theorem 2.1. Theorem 2.2. The convolutions Li xs ∗ xr and ln|1 − xs |x−1 ∗ xr exist and Li x ∗ s

xr

r r1 s 1 Li xs xr1 −1r−i Is,r−i xxi r 1 i0 r1 i

2.5


5

for r 0, 1, 2, . . . , s 1, 2, . . . and ln|1 − x

s

|x−1

∗

xr

r−1 r

1 −1r−i Is,r−i−1 xxi − Li xs xr s i

i0

2.6

for r, s 1, 2, . . . . Proof. It is obvious that Li xs ∗ xr 0 if x < 0. When x > 0, we have Li x ∗ s

xr

−s

x

x − t

r

0

−s

x

t

u−1 ln|1 − us |du dt

0

u−1 ln|1 − us |

0

x

x − tr dt du

u

x r1 r1 s r−i i ur−i ln|1 − us |du −1 x r 1 i0 i 0

r r1 1 r1 s x Lixs −1r−i xi Is,r−i x r 1 i0 r 1 i

2.7

proving 2.5. Next, using 1.8 and 2.5, we have −s ln|1 − xs |x−1 ∗ xr rLi xs ∗ xr−1 r−1 r s −1r−i−1 Is,r−i−1 xxi Li xs xr , i i0

2.8

and 2.6 follows. Corollary 2.3. The convolutions Li− xs ∗ x−r and ln|1 − xs |x−−1 ∗ x−r exist and Li− x ∗ s

x−r

r r1 s 1 Li− xs x−r1 −1r−i1 Is,r−i xx−i r 1 i0 r 1 i

2.9

for r 0, 1, 2, . . . , s 1, 2, . . . and ln|1 − x

s

|x−−1

∗

x−r

r−1 r i0

for r, s 1, 2, . . . .

1 −1r−i1 Is,r−i−1 xx−i − Li− xs x−r s i

2.10

6


Proof. Equations 2.9 and 2.10 are obtained applying a similar procedure as used in obtaining 2.5 and 2.6. The next two theorems were proved in 6 and to prove it, our set of negligible functions was extended to include finite linear sums of the functions ns Linr for s 0, 1, 2, . . . and r 1, 2, . . . . Theorem 2.4. The convolution Li x xr exists and

r r1 1 −1r−i xi Li x x r 1 i0 i r − i 12 r

2.11

for r 0, 1, 2, . . . . Theorem 2.5. The convolution ln|1 − x|x−1 xr exists and ln|1 −

x|x−1

x r

r−1 r −1r−i1 i0

i

r − i2

xi

2.12

for r 1, 2, . . . . Before proving some further results, we need the following lemma. Lemma 2.6. If r 1/s ∈ N for r, s 1, 2, . . . , then N − limIs,r n − n→∞

s r 12

.

2.13

Proof. Because

Is,r n

ns 1 − tr1/s 1 r1 1 n − 1 ln|1 − ns | − dt r1 r1 0 1−t

2.14

when r 1/s ∈ N, we have

Is,r n

r1/s−1 nsis 1 r1 1 n − 1 ln|1 − ns | − r1 r 1 i0 i 1

r1/s−1 nsis 1 r1 1 n − 1 s ln n ln1 − n−s − , r1 r 1 i0 i 1

and 2.13 follows. We now prove the following generalization of Theorems 2.4 and 2.5.

2.15


7

Theorem 2.7. The neutrix convolution Li xs xr exists when r 1/s ∈ N and

r r1 s2 −1r−i Li x x xi r 1 i0 i r − i 12 s

r

2.16

for r, s 1, 2, . . . . Proof. We put Li xs n Li xs τn x. Then the convolution Li xs n ∗ xr exists by Definition 1.1 and Li x n ∗ x s

r

n

r

Lit x − t dt s

nn−n

τn tLits x − tr dt

n

2.17

ln|1 − us | du dt u 0 0 n ln|1 − us | n −s x − tr dt du u 0 u

n r1 r1 s r−i i u−1 ln|1 − us | ur−i1 − nr−i1 du −1 x r 1 i0 i 0

r r1 s 1 r1 x Lins −1r−i xi Is,r−i n r 1 i0 r 1 i

2.18

0

I1 I2 , where I1

n 0

−s

Lits x − tr dt n

x − t

r

t

r r1 1 −1r−i xi Lins nr−i1 . r 1 i0 i Thus, using Lemma 2.6, we have

r r 1 −1r−i1 i s2 N − limI1 x. n→∞ r 1 i0 i r − i 12

2.19

Further, it is easily seen that nn−n lim

n→∞

τn tLits x − tr dt 0,

n

and 2.16 follows from 2.17, 2.19, and 2.20, proving the theorem.

2.20

8


Theorem 2.8. The neutrix convolution ln|1 − xs |x−1 xr exists when r 1/s ∈ N and ln|1 − x

s

|x−1

r−1 r −1r−i i x s x i r − i2 i0 r

2.21

for r, s 1, 2, . . . . Proof. Using Theorems 1.5 and 1.6, we have −s ln|1 − xs |x−1 xr rLi xs xr−1 N − lim Li xs τn x ∗ xr , n→∞

2.22

where, on integration by parts, we have

Li xs τn x ∗ xr

nn−n n

τn tLits x − tr dt r

−Lin x − n − s s

nn−n

ln|1 − ts |t−1 x − tr τn tdt

2.23

n

r

nn−n

Lits x − tr−1 τn tdt.

n

It is clear that nn−n lim

n→∞

lim

ln|1 − ts |t−1 x − tr τn tdt 0

n

nn−n

n→∞ n

Lit x − t s

r−1

2.24

τn tdt 0 .

It now follows from 2.23 and 2.24 that N − lim Li xs τn x ∗ xr 0. n→∞

2.25

Equation 2.21 now follows directly from 2.16 and 2.22, proving the theorem. Corollary 2.9. The neutrix convolutions Li− xs xr and ln |1 − xs |x−−1 xr exist when r 1/s ∈ N and

r r 1 −1r−i1 i s2 x− , Li− x x r 1 i0 i r − i 12 r−1 r −1r−i1 i x− ln|1 − xs |x−−1 xr s i0 i r − i2 s

for r, s 1, 2, . . .

r

2.26


9

Proof. Equations 2.26 are obtained applying a similar procedure as used in obtaining 2.16 and 2.21. Corollary 2.10. The neutrix convolutions Li xs x−r and Li− xs xr exist when r 1/s ∈ N and

Li x s

x−r

r r1 s −1i s − r − i 12 Is,r−i x xi 2 r 1 i0 i r − i 1

−1r1 Li xs xr1 , r1

r r1 s −1i1 2 s r x−i Li− x x s − − i 1 I r x s,r−i r 1 i0 i r − i 12

2.27

−1r1 Li− xs xr1 r1

for r, s 1, 2, . . . . Proof. Since the neutrix convolution product is distributive with respect to addition, we have Li xs xr Li xs ∗ xr −1r Li xs x−r ,

2.28

and 2.27 follows from 2.16 and 2.5. Equation 27 is obtained applying similar procedure as in the case of 2.27. Corollary 2.11. The neutrix convolutions ln|1 − xs |x−1 x−r and ln|1 − xs |x−−1 xr exist when r 1/s ∈ N and

ln|1 − x

s

|x−1

x−r

r−1 r −1i i0

i

r − i2

s − r − i2 Is,r−i−1 x xi

−1r Li xs xr , s r−1 r −1i1 2 s −1 r x−i ln|1 − x |x− x s − − i I r x s,r−i−1 2 i − i r i0

2.29

−1r Li− xs xr s

for r, s 1, 2, . . . Proof. Equation 2.29 follows from 2.21 and 2.6. Equation 29 is obtained applying similar procedure as in the case of 2.29.

10


Acknowledgment This research was supported by FEIT, University of Ss. Cyril and Methodius in Skopje, Republic of Macedonia, Project no. 08-3619/8.

References 1 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, NY, USA, 9th edition, 1972. 2 I. M. Gel’fand and G. E. Shilov, Generalized Functions, vol. 1, chapter 1, Academic Press, New York, NY, USA, 1964. 3 D. S. Jones, “The convolution of generalized functions,” The Quarterly Journal of Mathematics, vol. 24, pp. 145–163, 1973. 4 B. Fisher, “Neutrices and the convolution of distributions,” Univerzitet u Novom Sadu. Zbornik Radova Prirodno-Matematiˇckog Fakulteta. Serija za Matemati, vol. 17, no. 1, pp. 119–135, 1987. 5 J. G. van der Corput, “Introduction to the neutrix calculus,” Journal d’Analyse Math´ematique, vol. 7, pp. 291–398, 1959. ¨ ¸ ag, 6 B. Jolevska-Tuneska, B. Fisher, and E. Ozc ˘ “On the dilogarithm integral,” International Journal of Applied Mathematics, vol. 24, no. 3, pp. 361–369, 2011.

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